Resilience for stochastic systems interacting via a quasi-degenerate network

Abstract

A stochastic reaction-diffusion model is studied on a networked support. In each patch of the network, two species are assumed to interact following a non-normal reaction scheme. When the interaction unit is replicated on a directed linear lattice, noise gets amplified via a selfconsistent process,whichwe trace back to the degenerate spectrum of the embedding support. The same phenomenon holdswhen the system is bound to explore a quasidegenerate network. In this case, the eigenvalues of the Laplacian operator,which governs species diffusion, accumulate over a limited portion of the complex plane. The larger the network, the more pronounced the amplification. Beyond a critical network size, a system deemed deterministically stable, hence resilient, can develop seemingly regular patterns in the concentration amount. Non-normality and quasidegenerate networks may, therefore, amplify the inherent stochasticity and so contribute to altering the perception of resilience, as quantified via conventional deterministic methods.